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Fermat's Last Theorem

The Proof

Takeshi Saito

Translations of

MATHEMATICAL

American Mathematical Society

MONOGRAPHSVolume 245

Fermat's Last TheoremThe Proof

Fermat's Last TheoremThe Proof

Takeshi Saito

Translated from the Japanese by Masato Kuwata

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S Translations of

MATHEMATICALMONOGRAPHS

Volume 245

American Mathematical Society Providence, Rhode Island

10.1090/mmono/245

FERUMA YOSO (Fermat Conjecture)

by Takeshi Saito

c© 2009 by Takeshi SaitoFirst published 2009 by Iwanami Shoten, Publishers, Tokyo.

This English language edition published in 2014by the American Mathematical Society, Providence

by arrangement with the author c/o Iwanami Shoten, Publishers, Tokyo

Translated from the Japanese by Masato Kuwata

2010 Mathematics Subject Classification. Primary 11D41;Secondary 11F11, 11F80, 11G05, 11G18.

Library of Congress Cataloging-in-Publication Data

ISBN 978-0-8218-9849-9 Fermat’s last theorem: the proof(Translations of mathematical monographs ; volume 245)

The first volume was catalogued as follows:Saito, Takeshi, 1961–

Fermat’s last theorem: basic tools / Takeshi Saito ; translated by MasatoKuwata.—English language edition.

pages cm.—(Translations of mathematical monographs ; volume 243)First published by Iwanami Shoten, Publishers, Tokyo, 2009.Includes bibliographical references and index.ISBN 978-0-8218-9848-2 (alk. paper)1. Fermat’s last theorem. 2. Number theory. 3. Algebraic number theory.

I. Title. II. Title: Fermat’s last theorem: basic tools.

QA244.S2513 2013512.7’4–dc23

2013023932

c© 2014 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rightsexcept those granted to the United States Government.

Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Information on copying and reprinting can be found in the back of this volume.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 19 18 17 16 15 14

Contents

Basic Tools

Preface xi

Preface to the English Edition xvii

Chapter 0. Synopsis 10.1. Simple paraphrase 10.2. Elliptic curves 30.3. Elliptic curves and modular forms 50.4. Conductor of an elliptic curve and level of a modular

form 70.5. �-torsion points of elliptic curves and modular forms 9

Chapter 1. Elliptic curves 131.1. Elliptic curves over a field 131.2. Reduction mod p 151.3. Morphisms and the Tate modules 221.4. Elliptic curves over an arbitrary scheme 261.5. Generalized elliptic curves 29

Chapter 2. Modular forms 352.1. The j-invariant 352.2. Moduli spaces 372.3. Modular curves and modular forms 402.4. Construction of modular curves 442.5. The genus formula 522.6. The Hecke operators 552.7. The q-expansions 582.8. Primary forms, primitive forms 622.9. Elliptic curves and modular forms 65

v

vi CONTENTS

2.10. Primary forms, primitive forms, and Hecke algebras 662.11. The analytic expression 702.12. The q-expansion and analytic expression 742.13. The q-expansion and Hecke operators 77

Chapter 3. Galois representations 813.1. Frobenius substitutions 823.2. Galois representations and finite group schemes 863.3. The Tate module of an elliptic curve 893.4. Modular �-adic representations 913.5. Ramification conditions 963.6. Finite flat group schemes 1003.7. Ramification of the Tate module of an elliptic curve 1033.8. Level of modular forms and ramification 108

Chapter 4. The 3–5 trick 1114.1. Proof of Theorem 2.54 1114.2. Summary of the Proof of Theorem 0.1 116

Chapter 5. R = T 1195.1. What is R = T? 1195.2. Deformation rings 1225.3. Hecke algebras 1265.4. Some commutative algebra 1315.5. Hecke modules 1355.6. Outline of the Proof of Theorem 5.22 137

Chapter 6. Commutative algebra 1436.1. Proof of Theorem 5.25 1436.2. Proof of Theorem 5.27 149

Chapter 7. Deformation rings 1597.1. Functors and their representations 1597.2. The existence theorem 1617.3. Proof of Theorem 5.8 1627.4. Proof of Theorem 7.7 166

Appendix A. Supplements to scheme theory 171A.1. Various properties of schemes 171A.2. Group schemes 175A.3. Quotient by a finite group 177A.4. Flat covering 178

CONTENTS vii

A.5. G-torsor 179A.6. Closed condition 182A.7. Cartier divisor 183A.8. Smooth commutative group scheme 185

Bibliography 189

Symbol Index 197

Subject Index 199

The Proof

Preface ix

Preface to the English Edition xv

Chapter 8. Modular curves over Z 18.1. Elliptic curves in characteristic p > 0 18.2. Cyclic group schemes 68.3. Drinfeld level structure 128.4. Modular curves over Z 208.5. Modular curve Y (r)Z[ 1r ]

25

8.6. Igusa curves 328.7. Modular curve Y1(N)Z 378.8. Modular curve Y0(N)Z 418.9. Compactifications 48

Chapter 9. Modular forms and Galois representations 619.1. Hecke algebras with Z coefficients 619.2. Congruence relations 709.3. Modular mod � representations and non-Eisenstein

ideals 769.4. Level of modular forms and ramification of �-adic

representations 819.5. Old part 909.6. Neron model of the Jacobian J0(Mp) 979.7. Level of modular forms and ramification of mod �

representations 102

viii CONTENTS

Chapter 10. Hecke modules 10710.1. Full Hecke algebras 10810.2. Hecke modules 11310.3. Proof of Proposition 10.11 11810.4. Deformation rings and group rings 12510.5. Family of liftings 12910.6. Proof of Proposition 10.37 13610.7. Proof of Theorem 5.22 140

Chapter 11. Selmer groups 14311.1. Cohomology of groups 14311.2. Galois cohomology 14911.3. Selmer groups 15711.4. Selmer groups and deformation rings 16111.5. Calculation of local conditions and proof of

Proposition 11.38 16511.6. Proof of Theorem 11.37 169

Appendix B. Curves over discrete valuation rings 179B.1. Curves 179B.2. Semistable curve over a discrete valuation ring 182B.3. Dual chain complex of curves over a discrete valuation

ring 187

Appendix C. Finite commutative group scheme over Zp 191C.1. Finite flat commutative group scheme over Fp 191C.2. Finite flat commutative group scheme over Zp 192

Appendix D. Jacobian of a curve and its Neron model 199D.1. The divisor class group of a curve 199D.2. The Jacobian of a curve 201D.3. The Neron model of an abelian variety 205D.4. The Neron model of the Jacobian of a curve 209

Bibliography 213

Symbol Index 217

Subject Index 221

Preface

It has been more than 350 years since Pierre de Fermat wrote inthe margin of his copy of Arithmetica of Diophantus:

It is impossible to separate a cube into two cubes, or abiquadrate into two biquadrates, or in general any powerhigher than the second into powers of like degree; I havediscovered a truly remarkable proof which this margin istoo small to contain.1

This is what we call Fermat’s Last Theorem. It is certain that hehas a proof in the case of cubes and biquadrates (i.e., fourth pow-ers), but it is now widely believed that he did not have a proof inthe higher degree cases. After enormous effort made by a great num-ber of mathematicians, Fermat’s Last Theorem was finally proved byAndrew Wiles and Richard Taylor in 1994.

The purpose of this book is to give a comprehensive account ofthe proof of Fermat’s Last Theorem. Although Wiles’s proof is basedon very natural ideas, its framework is quite complex, some partsof it are very technical, and it employs many different notions inmathematics. In this book I included parts that explain the outline ofwhat follows before introducing new notions or formulating the proofformally. Chapter 0 and §§5.1, 5.5, and 5.6 in Chapter 5 are thoseparts. Logically speaking, these are not necessary, but I includedthese in order to promote better understanding. Despite the aim ofthis book, I could not prove every single proposition and theorem.For the omitted proofs please consult the references indicated at theend of the book.

The content of this book is as follows. We first describe the roughoutline of the proof. We relate Fermat’s Last Theorem with elliptic

1Written originally in Latin. English translation is taken from Dickson, L. E.,History of the theory of numbers. Vol. II: Diophantine analysis, Chelsea Publish-ing Co., New York, 1966.

ix

x PREFACE

curves, modular forms, and Galois representations. Using these rela-tions, we reduce Fermat’s Last Theorem to the modularity of certain�-adic representations (Theorem 3.36) and a theorem on the level ofmod � representations (Theorem 3.55). Next, we introduce the no-tions of deformation rings and Hecke algebras, which are incarnationsof Galois representations and modular forms, respectively. We thenprove two theorems on commutative algebra. Using these theorems,we reduce Theorem 3.36 to certain properties of Selmer groups andHecke modules, which are also incarnations of Galois representationsand modular forms.

We then construct fundamental objects, modular curves over Z,and the Galois representations associated with modular forms. Thelatter lie in the foundation of the entire proof. We also show a partof the proof of Theorem 3.55. Finally, we define the Hecke modulesand the Selmer groups, and we prove Theorem 3.36, which completesthe proof of Fermat’s Last Theorem.

The content of each chapter is summarized at its beginning, butwe introduce them here briefly. In Chapter 0,∗ we show that Fer-mat’s Last Theorem is derived from Theorem 0.13, which is aboutthe connection between elliptic curves and modular forms, and Theo-rem 0.15, which is about the ramification and level of �-torsion pointsof an elliptic curve. The objective of Chapters 1–4∗ is to understandthe content of Chapter 0 more precisely. The precise formulations ofTheorems 0.13 and 0.15 will be given in Chapters 1–3. In the proofpresented in Chapter 0, the leading roles are played by elliptic curves,modular forms, and Galois representations, each of which will be themain theme of Chapters 1, 2, and 3. In Chapter 3, the modularity of�-adic representations will be formulated in Theorem 3.36. In Chap-ter 4, using Theorem 4.4 on the rational points of an elliptic curve,we deduce Theorem 0.13 from Theorem 3.36. In §4.2, we review theoutline of the proof of Theorem 0.1 again.

In Chapters 5–7,∗ we describe the proof of Theorem 3.36. Theprincipal actors in this proof are deformation rings and Hecke alge-bras. The roles of these rings will be explained in §5.1. In Chapter 5,using two theorems of commutative algebra, we deduce Theorem 3.36from Theorems 5.32, 5.34, and Proposition 5.33, which concern theproperties of Selmer groups and Hecke modules. The two theorems

∗Chapters 0–7 along with Appendix A appeared in Fermat’s Last Theorem:Basic Tools, a translation of the Japanese original.

PREFACE xi

in commutative algebra will be proved in Chapter 6. In Chapter 7,we will prove the existence theorem of deformation rings.

In Chapter 8, we will define modular curves over Z and studytheir properties. Modular forms are defined in Chapter 2 using mod-ular curves over Q, but their arithmetic properties are often derivedfrom the behavior of modular curves over Z at each prime number.Modular curves are known to have good reduction at primes not divid-ing their levels, but it is particularly important to know their preciseproperties at the prime factors of the level. A major factor that madeit possible to prove Fermat’s Last Theorem within the twentieth cen-tury is that properties of modular curves over Z had been studiedintensively. We hope the reader will appreciate this fact.

In Chapter 9, we construct Galois representations associated withmodular forms, using the results of Chapter 8, and prove a part ofTheorem 3.55 which describes the relation between ramification andthe level. Unfortunately, however, we could not describe the cele-brated proof of Theorem 3.55 in the case of p ≡ 1 mod � by K. Ri-bet because it requires heavy preparations, such as the p-adic uni-formization of Shimura curves and the Jacquet–Langlands–Shimizucorrespondence of automorphic representations.

In Chapter 10, using results of Chapters 8 and 9, we constructHecke modules as the completion of the singular homology groups ofmodular curves, and we then prove Theorem 5.32(2) and Proposi-tion 5.33. In Chapter 11, we introduce the Galois cohomology groupsand define the Selmer groups. Then we prove Theorems 5.32(1)and 5.34. The first half of Chapter 11 up to §11.3 may be read inde-pendently as an introduction to Galois cohomology and the Selmergroups.

Throughout the book, we assume general background in numbertheory, commutative algebra, and general theory of schemes. Theseare treated in other volumes in the Iwanami series: Number Theory 1 ,2, and 3, Commutative algebras and fields (no English translation),and Algebraic Geometry 1 and 2. For scheme theory, we give a briefsupplement in Appendix A after Chapter 7. Other prerequisites aresummarized in Appendices B, C, and D at the end of the volume.

In Appendix B, we describe algebraic curves over a discrete valua-tion rings and semistable curves in particular, as an algebro-geometricpreparation to the study of modular curves over Z. In Appendix C,we give a linear algebraic description of finite flat commutative groupschemes over Zp, which will be important for the study of p-adic

xii PREFACE

Galois representations of p-adic fields. Finally, in Appendix D, wegive a summary on the Jacobian of algebraic curves and its Neronmodel, which are indispensable to study the Galois representationsassociated with modular forms.

If we gave a proof of every single theorem or proposition in Chap-ters 1 and 2, it would become a whole book by itself. So, we onlygive proofs of important or simple properties. Please consider thesechapters as a summary of known facts. Reading the chapters on el-liptic curves and modular forms in Number Theory 1 ,2, and 3 wouldalso be useful to the reader.

At the end of the book, we give references for the theorems andpropositions for which we could not give proofs in the main text. Theinterested reader can consult them for further information. We regretthat we did not have room to mention the history of Fermat’s LastTheorem. The reader can also refer to references at the end of thebook. Due to the nature of this book, we did not cite the originalpaper of each theorem or proposition, and we beg the original authorsfor mercy.

I would be extremely gratified if more people could appreciate oneof the highest achievements of the twentieth century in mathematics.I would like to express sincere gratitude to Professor Kazuya Katofor proposing that I write this book. I would also thank MasatoKurihara, Masato Kuwata, and Kazuhiro Fujiwara for useful advice.Also, particularly useful were the survey articles [4], [5], and [24].I express here special thanks to their authors.

This book was based on lectures and talks at various places, in-cluding the lecture course at the University of Tokyo in the first se-mester of 1996, and intensive lecture courses at Tohoku University inMay 1996, at Kanazawa University in September 1996, and at NagoyaUniversity in May 1999. I would like to thank all those who attendedthese lectures and took notes. I would also like to thank formerand current graduate students at the University of Tokyo, KeisukeArai, Shin Hattori, and Naoki Imai, who read the earlier manuscriptcarefully and pointed out many mistakes. Most of the chapters up toChapter 7 were written while I stayed at Universite Paris-Nord, Max-Planck-Institut fur Mathematik, and Universitat Essen. I would liketo thank these universities and the Institute for their hospitality andfor giving me an excellent working environment.

PREFACE xiii

This book is the combined edition of the two books in the Iwanamiseries The Development of Modern Mathematics: Fermat’s Last The-orem 1 first published in March 2000 and containing up to Chapter 7;and Fermat’s Last Theorem 2 published in February 2008.

Since 1994 when the proof was first published, the development ofthis subject has been remarkable: Conjecture 3.27 has been proved,and Conjecture 3.37 has almost been proved. Also, Theorem 5.22 hasbeen generalized widely, and its proof has been simplified greatly. Weshould have rewritten many parts of this book to include recent de-velopments, but we decided to wait until another opportunity arises.

On the occasion of the second edition, we made corrections toknown errors. However, we believe there still remain many mistakesyet to be discovered. I apologize in advance, and would be grateful ifthe reader could inform me.

Takeshi SaitoTokyo, Japan

November 2008

Preface to the English Edition

This is the second half of the English translation of Fermat’s LastTheorem in the Iwanami series, The Development of Modern Math-ematics. Though the translation is based on the second combinededition of the original Japanese book published in 2008, it will bepublished in two volumes. The first volume, Fermat’s Last Theorem:Basic Tools , contains Chapters 0–7 and Appendix A. The second vol-ume, Fermat’s Last Theorem: The Proof , contains Chapters 8–11 andAppendices B, C, and D.

This second volume of the book on the proof of Fermat’s LastTheorem by Wiles and Taylor presents a full account of the proofstarted in the first volume. As well as the proof itself, basic materialsbehind the proof, including the Galois representations associated withmodular forms, the integral models of modular curves, the Heckemodules, the Selmer groups, etc., are studied in detail.

The author hopes that, through this edition, a wider audience ofreaders will appreciate one of the deepest achievements of the twen-tieth century in mathematics.

My special thanks are due to Dr. Masato Kuwata, who not onlytranslated the Japanese edition into English but also suggested manyimprovements in the text so that the present English edition is morereadable than the original Japanese edition.

Takeshi SaitoTokyo, JapanOctober 2014

xv

Bibliography

References for theorems and propositionsthat were not proved in the text

Chapter 8

[1] J. Silverman, The Arithmetic of Elliptic Curves, Graduate Textsin Math., Springer, 106, 1986.

[2] P. Deligne, M. Rapoport, Les schemas de modules de courbes el-liptiques, in Modular Functions of One Variable II, Lecture Notesin Math., Springer, 349, 1973, 143–316.

[3] N. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves, Annalsof Math. Studies, Princeton Univ. Press, 151, 1994.

[4] H. Hida, Geometric modular forms and elliptic curves, World Sci-entific, 2000.

[5] B. J. Birch, W. Kuyk (eds.), Modular Functions of One Vari-able IV, Lecture Notes in Math., Springer, 476, 1973.

� Lemma 8.37: [2] III Corollaire 2.9, p.211, [3] Corollary 4.7.2.� Lemma 8.41: p = 2, 3: [1] Appendix A, Proposition 1.2 (c).� Example 8.65: [5] Table 6, p.143.� Proposition 8.69: [2] VII Costruction 1.15, p.297.� Theorem 8.77: [2] V Theoreme 2.12, [3] Theorem 13.11.4.

Chapter 9

[6] H. Carayol, Sur les representations galoisiennes modulo � at-tachees aux formes modulaires, Duke Math. J. 59 (1989), 785–801.

[7] T. Miyake, Modular forms. Translated from the Japanese byYoshitaka Maeda. Springer-Verlag, Berlin, 1989. x+335 pp.

213

214 BIBLIOGRAPHY

[8] K. Ribet, On modular representations of Gal(Q/Q) arising frommodular forms, Inventiones Math., 100 (1990), 431–476.

� Theorem 3.55(2) (ii) ⇒ (i): [6].� Theorem 9.40: [7] Corollary 4.6.20.� Theorem 3.55(1) (ii) ⇒ (i) the case p ≡ 1 mod �: [8].

Chapter 10

[9] J.-P. Serre, Arbres, amalgames, SL2, Asterisque 46, SocieteMathematique de France, Paris, 1977.

[10] , Le probleme des groupes de congruence pour SL2, Ann.of Math. 92 (1970), 489–527.

[11] L. E. Dickson, Linear groups with an exposition of the Galoisfield theory, Teubner, Leipzig, 1901.

� Theorem 10.15(1): [9] Chapitre II 1.4 Theoreme 3 p.110.� Theorem 10.15(2): [10] 2.6 Corollaire 3 p.449 (If we let K = Q,

S = {p,∞}, and q = (N), then we have Γq = Γ(N) and Eq =

E(N).)� Theorem 10.28: [11] sections 255, 260.

Chapter 11

[12] J.-P. Serre, Corps Locaux, 3e ed., Hermann, Paris, 1980.[13] , Cohomologie galoisienne, 5e ed., Lecture Notes in Math.,

Springer-Verlag, Berlin, 5, 1994.[14] J. S. Milne, Arithmetic duality theorems, Perspectives in Math. 1,

Academic Press, Boston, 1986.

� General theory of Galois cohomology and duality theorem: [4].� Proposition 11.11(1): [12] Chapitre X §3 b), (2): ibid., Propo-sition 9.� Proposition 11.18: [14] Corollary 2.3.� Proposition 11.20: [14] Theorem 2.8.� Proposition 11.25(1): [14] Corollary 4.15.� Proposition 11.25(2): [14] Theorem 4.10.� Proposition 11.27: [14] Theorem 5.1.

BIBLIOGRAPHY 215

Appendix B

[65] P. Deligne, N. Katz, Groupes de Monodromie en GeometrieAlgebrique (SGA 7) II, Lecture Notes in Math., Springer, 340,(1973).

Lemma B.4 (iii) ⇒ (i) The case where k is general: [65] Exp. XV,Theoreme 1.2.6.Lemma B.12: [65] Exp. X, Corollaire 1.8.

Appendix C

[15] J.-M. Fontaine, Groupes p-divisible sur les corps locaux,Asterisque 47–48, Soc. Math. de France, 1977.

[16] J.-M. Fontaine, G. Laffaille, Construction de representations p-

adiques, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 547–608(1983).

[17] N. Wach, Representations cristallines de torsion, CompositioMath. 108 (1997), 185–240.

� Theorem C.1:[15] Chapitre III.� Theorem C.6:[16], [17].

Appendix D

[18] S. Mukai, An introduction to invariants and moduli, Translatedfrom the 1998 and 2000 Japanese editions by W. M. Oxbury.Cambridge Studies in Advanced Mathematics, 81. CambridgeUniversity Press, Cambridge, 2003. xx+503 pp.

[19] M. Artin, Neron models, in G. Cornell, J. Silverman, (eds.),Arithmetic Geometry, Springer, 1986 pp.213–230.

[20] J. S. Milne, Jacobian varieties, in G. Cornell, J. Silverman, (eds.),Arithmetic Geometry, Springer, 1986 pp.167–212.

[21] S. Bosch, W. Lutkebohmert, M. Raynaud, Neron models,Springer, 1990.

[22] M. Raynaud, Jacobienne des courbes modulaires et operateursde Hecke, in “Courbes modulaires et courbes de Shimura”,Asterisque 196–197, Soc. Math. de France, 1991, 9–25.

216 BIBLIOGRAPHY

[23] A. Grothendieck, Modeles de Neron et monodromie, in Groupesde Monodromie en Geometrie Algebrique, SGA 7I, Lecture Notesin Math., Springer, 288, 1972, 313–523. Proposition 5.13.

� Jacobians and Neron models: [19], [20], [21]� Curves and their Jacobians: [18] Chapter 8.� Abel’s theorem: [18] Theorem 9.8.6.� Theorem D.3: [21] Theorems 8.4/3 and 9.3/1.� Theorem D.7: [21] Theorem 8.2/3 and Propositions 9.2/5and 10.� Theorem D.8: [21] Corollary 1.3/1.� Proposition D.12: [22], Proposition 6.� Theorem D.17(1) � = p: [21] Theorem 7.4/5 (b)⇔(d).� Theorem D.17(2) � = p: [21] Theorem 7.4/6.� Theorem D.17(2) � = p: [23] Proposition 5.13.� Theorem D.19(1): [21] Theorem 9.5/4, p.267.� Theorem D.19(2): [21] Theorem 9.6/1, p.274.

Symbol Index

∪, 1491-cocycle, 144

αq, 125〈a〉, 22〈a〉−1/2, 132Af , 111, 131AΣ′ , 113

Br(F ), 150

nBr(F ), 150

Cq(G,M), 149C(X,F), 137

ΔQ, 126Δq, 125Δp, 117DefR-G(M), 146Def0R-G, 146Def ρ,DΣ

(O/(πn)[ε])[Vn], 163div f , 199D(G), 191, 194D(ρ), 108D(X), 199

ε, 70εf , 70e, 51ex, 24E(N), 118

E(N), 118E(pe), 2Eqrr , 51

End0R(M), 147ExtR-G(M,N), 148

fa, 65

f �Q, 132F , 2, 191, 193Fe, 2FS , 1FΣ′,Σ, 116

Γ0,∗(p,N), 118Γ(N), 118Γ1(N), 38

Γ(N), 118Γ(r), 29G-coinvariant, 144G×, 10G(a,b), 33GF , 143GS , 155Grass(OE[N ], N), 41

H0(G,M), 144H1(G,M), 144H1

f (Q�,Wn), 161

H1f (Qp,M), 151

H1f (Qp,HomR(M,N)), 152

217

218 SYMBOL INDEX

H1f (Qp, N), 152

H1s (Q�,Wn), 162

Hq(F,M), 150Hq(G,M), 149

iΣ, 110I0(N,n), 62I1(N,n), 66I0(N,n), 62Ig(Mpa, r)Fp

, 33

Ig(Mpa, r)P=0Fp

, 34

Ig(Mpa, r)Fp, 56

j, 21, 29ja, 35, 42, 55, 56j0, j1, 49, 58, 59, 118J0,1(N,M), 70J0(N)Q, 61J1(N)Q, 66

Kf , 63

λ, 31�-divisible group, 208(Lp)p∈S , 157(L∨

p )p∈S , 158LΣ, 162Lift0R-G, 146LiftR-G(M), 146Liftρ,DΣ

(O/(πn)[ε])Vn, 163

μ, 25μ×N , 12

m�Q, 130

m�∅,Q, 135

m∗Σ′,Σ, 114

mΣ′,Σ, 115m′

Q, 131mRΣ

, 163mΣ, 110m′

Σ, 110

M , 146

Mρ, 146M0(N)E , 41M1(N)E , 15M(1), 150MG, 144MG, 144

M �Q, 134

MΣ, 114M∨, 150M, 21M0,∗(N, r)Z[ 1r ]

, 25

M0(N), 20M0(N)E , 13M0(N)Fp

, 23M1,∗(N, r)Z[ 1r ]

, 25

M1,0(M,N), 57M1(N), 20M1(N)E , 13M(r)Z[ 1r ]

, 25

NfL, 202N -IsogE , 41

ord, 2

ϕΣ, 108Φ, 97pRΣ

, 163〈P 〉, 13Pf,p(U), 111, 131Pp(U), 94, 109Pq(U), 130PX/S(T ), 202Pic(X), 199, 201

Pic0(X), 202

Pic0X/S , 203PicX/k, 202

SYMBOL INDEX 219

Q, 126Q, 126

Q, 125QS , 155

R�Q, 135

RΣ, 114R[ε], 146

s0, s1, 118sd, 45, 59sk, 82sk(q), 50S0(N), 61S1(N), 66Sss, 2Sel(E, n), 158SelL(M), 157SelΣ(Wn), 162

tk, 82t∅,Q, 129

t�Σ′,Σ, 113

t�∅,Q, 130

T (N)Z, 108T (NΣ)

′O, 110

T [N ], 16T (N), 16T�J0(N), 70T0(N)Z, 62T1(N)Z, 67TN,E , 41

T �Σ, 110

T �Q, 130

T �� , 113

T �q , 129

T , 97T ′, 103Tor, 137

U�, 115Up, 92, 115

V , 2, 72, 161, 191V 0n , 161

Ve, 2Vf , 71V�J0(N), 70

w2N ′ , 22

W , 161W 0

n , 162

X0(N)Z, 48X0,∗(N, r)Z[ 1r ]

, 54

X1(Mp)balZ[ζp], 59

X1(N)Z, 48X1,∗(Mp, r)bal

Z[ 1r ,ζp], 58

X1,∗(N, r)Z[ 1r ], 54

X1,0(M,N)Z, 58X1,0(N,M)Z, 58

X(p), 1X(1)Z |∧∞, 51

Y0,∗(N, r)Z[ 1r ], 42

Y0(N)Z, 23Y1(4), 20, 47Y1,∗(N, r)Z[ 1r ]

, 31

Y1,0(M,N)Z, 57Y1(N)an, 38Y1(N)Z, 24Y (1)Z, 29Y (2), 31Y (3), 25Y (r)an, 29Y (r)Z[ 1r ]

, 26

Z1(G,M), 144

Z, 144Z((q)), 51

Subject Index

absolute Frobenius morphism,1

annihilator, 153Atkin–Lehner involution, 23

Brauer group, 150

characterof f , 70

congruence relation, 73connected component, 205cup product, 149curve, 179cyclic group scheme, 9

diamond operator, 22, 67Dieudonne module, 191divisor, 199divisor class group, 199divisor group, 199Drinfeld level structure, 13dual chain complex, 182dual local condition, 158

Eisenstein ideal, 77etalefiltered ϕ-module, 192

exact order N , 13extension, 147, 196

filtered ϕ-module, 192

finite G-module, 143finite R-G-module, 143Frobenius morphism, 23full Hecke algebra, 110, 130full set of sections, 6

G-coinvariant, 144generator, 9genus, 179good, 208good reduction, 182, 205

Hecke algebra, 62, 67Hecke module, 114, 134Hecke operator, 67

Igusa curve, 33index, 182infinitesimal deformation, 146infinitesimal lifting, 146

Jacobian, 203

�-divisible group, 208local condition, 157

minimal resolution ofsingularities, 188

multiplicativefiltered ϕ-module, 192

221

222 INDEX

Neron model, 205node, 180non-Eisenstein, 123non-Eisenstein ideal, 78normof an invertible sheaf, 202

old part, 911-cocycle, 144ordinary, 2

perfect complex, 136Petersson product, 65, 68Picard functor, 202Picard group, 201preserve the determinant, 146primary form, 70principal divisor, 199profinite group, 143

relative Frobenius morphism,2

right bounded, 136

scheme of generators, 10Selmer group, 157of an elliptic curve, 158

semistable, 182, 208semistable reduction, 182, 205singular chain complex, 137strongly divisible, 192supersingular, 2

Tate curve, 51Tate twist, 150

unramified part, 151

weakly semistable, 182

Selected Published Titles in This Series

245 Takeshi Saito, Fermat’s Last Theorem, 2014

243 Takeshi Saito, Fermat’s Last Theorem, 2013

242 Nobushige Kurokawa, Masato Kurihara, and Takeshi Saito,Number Theory 3, 2012

240 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number

Theory 2, 2011

235 Mikio Furuta, Index Theorem. 1, 2007

230 Akira Kono and Dai Tamaki, Generalized Cohomology, 2006

227 Takahiro Kawai and Yoshitsugu Takei, Algebraic Analysis ofSingular Perturbation Theory, 2005

224 Ichiro Shigekawa, Stochastic Analysis, 2004

218 Kenji Ueno, Algebraic Geometry 3, 2003

217 Masaki Kashiwara, D-modules and Microlocal Calculus, 2003

211 Takeo Ohsawa, Analysis of Several Complex Variables, 2002

210 Toshitake Kohno, Conformal Field Theory and Topology, 2002

209 Yasumasa Nishiura, Far-from-Equilibrium Dynamics, 2002

208 Yukio Matsumoto, An Introduction to Morse Theory, 2002

207 Ken’ichi Ohshika, Discrete Groups, 2002

206 Yuji Shimizu and Kenji Ueno, Advances in Moduli Theory, 2002

205 Seiki Nishikawa, Variational Problems in Geometry, 2002

201 Shigeyuki Morita, Geometry of Differential Forms, 2001

199 Shigeyuki Morita, Geometry of Characteristic Classes, 2001

197 Kenji Ueno, Algebraic Geometry 2, 2001

195 Minoru Wakimoto, Infinite-Dimensional Lie Algebras, 2001

189 Mitsuru Ikawa, Hyperbolic Partial Differential Equations and WavePhenomena, 2000

186 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, NumberTheory 1, 2000

185 Kenji Ueno, Algebraic Geometry 1, 1999

183 Hajime Sato, Algebraic Topology: An Intuitive Approach, 1999

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/mmonoseries/.

www.ams.orgAMS on the Web

MMONO/245

This is the second volume of the book on the proof of Fermat’s Last Theorem by Wiles and Taylor (the first volume is published in the same series; see MMONO/243). Here the detail of the proof announced in the first volume is fully exposed. The book also includes basic materials and constructions in number theory and arithmetic geometry that are used in the proof.

In the first volume the modularity lifting theorem on Galois repre-sentations has been reduced to properties of the deformation rings and the Hecke modules. The Hecke modules and the Selmer groups used to study deformation rings are constructed, and the required properties are established to complete the proof.

The reader can learn basics on the integral models of modular curves and their reductions modulo p that lay the foundation of the construction of the Galois representations associated with modular forms. More background materials, including Galois cohomology, curves over integer rings, the Néron models of their Jacobians, etc., are also explained in the text and in the appen-dices.

For additional information and updates on this book, visit

www.ams.org/bookpages/mmono-245